Mathematics and the Real World (58 page)

This distinction between the logical formulation of a mathematical result and the intuitive thinking carried out before the formulation of the result is also reflected in the way information is transmitted between mathematicians. A conversation on a mathematical subject is unlike mathematical writing that we recognize from our studies, books, or academic articles. We are often asked about the utility of flying for several hours to meet a colleague in a research project or to participate in a mathematics conference in some distant land. Would it not be enough, in this age of electronic communications, to read articles or correspond via e-mail? The answer is that in the intuitive thinking stage there is no substitute for face-to-face meetings. Information, which in a certain sense is subconscious and as such is difficult even to define, can be transmitted only in such a meeting. This is not confined to mathematics alone but applies to other areas of research too. My reason for stressing this aspect of mathematics is that it is a common misconception that logical mathematics will be less in need of the intuitive, subconscious level of thinking. The opposite is the case: most mathematical research is based on that intuition.

The differences we have described are responsible for the fact that unexplained events related to thinking, which also exist in other disciplines, occur frequently among mathematicians. It is not unusual that a colleague asks me to help him check a mathematical development he has conceived and insists on showing me the problem, despite my objections that I do not understand anything about the subject. He starts covering the blackboard in my office with the development in question, and after a while, without my having uttered a word, he stops, thinks, and says, “Thank you very much. Now I understand what has happened. You have helped me a great deal.” And I have indeed helped him a lot, just as I have been helped in a similar manner by colleagues. The very fact that we have tried to explain our findings to someone who is likely to have at least some idea of what we are trying to achieve is of great help.

Another well-known fact is that intuitive thinking to a large extent takes place subconsciously, sometimes even during sleep. Both Poincaré, whom we have discussed previously, and Jacques Hadamard, another leading French mathematician, wrote in their books about mathematical
problems that they tried to solve over a long period, and then, in the course of an event completely unrelated to mathematics, the complete solution suddenly dawned on them.

This phenomenon is known at all levels of the profession. Just recently I was deeply engrossed in trying to solve a problem but made hardly any significant progress in my daily work. One morning I woke up with the complete solution to the problem, or so I thought. Checking it in the office, I realized that the solution was incomplete. The complete solution required another night's sleep. The conclusion to be drawn is not to have a sound sleep every time you encounter a mathematical problem that you do not manage to solve. Investing effort and concentration are certainly essential. After you have invested time and much effort, having a break does not mean that your brain stops trying to solve the problem. Moreover, the break is likely to be beneficial.

Another aspect of subconscious thinking is the context in which the result of such thinking comes to light. I used to live about fifty minutes’ drive from my office in the Weizmann Institute. With a frequency that I cannot explain, at a certain point where the tree-lined road curved as I drove home, I would realize that I had made a mistake in my work that day. It would take a day or two to correct the error, but then, at the same spot on my journey home, another error would spring into my mind (obviously, when I moved closer to the office, the number of mistakes I had to correct in my articles declined significantly).

Colleagues tell me that quite often the solution to a problem, or a new idea leading to the solution, comes to mind while they are watching television (even today my wife finds it hard to believe that when I am watching a sitcom, I am actually working!). It may be that there is no significance in and no statistical explanation for these occasions, and the fact that we attribute importance to them is a result of some mental illusion. Such an illusion could derive from the fact that exceptional events, like discovering an error while driving past trees, are more memorable than more normal occurrences, such as discovering a mistake while working in the office. Yet it may also be that something in our brain causes solutions to spring to mind in just such unlikely situations.

Another surprising factor is the different degree of difficulty between solving a mathematical problem if we know that someone else has already solved it, say, as an exercise in a class, and solving a new problem that no one has solved yet. The latter are called open problems. The story is told of the mathematician John Milnor, that when he was a student he had dozed off in a class when the lecturer wrote on the board an open problem in mathematics. When he awoke from his nap and saw what was on the board, he thought that it was an assignment to be done for the following week. The next week he arrived with the solution. John Milnor is a top mathematician. In 1962, at the age of thirty-one, he won the Fields Medal (officially the International Medal for Outstanding Discoveries in Mathematics), awarded to mathematicians under the age of forty; in 1989 he won the Wolf Prize; and in 2011 he was the Abel Prize laureate. Of course you have to be a genius like John Milnor to solve an open problem as if it were simply an exercise to prepare for the next class, but this sort of thing is known to occur at all levels of research: you can work on an open problem for many months, and only after you have found the solution do you realize that the problem could have been solved much more simply, but that is being wise with the benefit of hindsight.

The fact that research problems, that is, open problems so far unsolved, are more difficult than other problems set in class frustrates many students setting out along the path of research. After investing great effort in solving an open problem, they discover that they could have solved it more simply, just as they had solved other problems in the various courses they took. Many of them draw the conclusion that they are not suited to research, which may be an incorrect conclusion. It is not clear why it is that the fact that a problem is an open problem makes it more difficult to solve. In my opinion, a reasonable explanation is that thinking by comparison and creative thinking may take place in different parts of the brain. The creative section of the brain works less efficiently than does the comparative part. When the brain has to solve a problem, it channels the problem to what it considers to be the appropriate section for handling the problem, and that is the cause for the difference in the level of difficulty in solving a problem presented as an exercise and one introduced as an open problem. I would
even go so far as to say that the capability of creative thinking is what differentiates man from the rest of the animal world, but that of course is pure speculation.

Does creative thinking in mathematics decline with age? This is the place to dispel a myth about the relation between mathematical research and age. According to the myth, after a certain age, some claim as early as thirty, creative ability in mathematics declines and eventually disappears. That is not so, and I will explain why. Physicists, and I have heard this from top physicists, agree that creative ability in physics declines from the age of thirty-something. That does not mean that after that age physicists do not contribute to their profession, but rather that the breakthroughs and innovative developments are achieved by the younger physicists. Yuval Ne'eman claimed that this does not refer to chronological age but to the length of time in the discipline (Ne’eman himself completed his doctorate at a relatively late age), and the reason for the reduction in the ability to introduce innovation is that as people accumulate experience and become accustomed to certain truths and to a certain method of research, it gets harder for them to challenge accepted norms and practices and to disprove or change them. And indeed, the beginning of most breakthroughs in physics was the refutation of deep-rooted beliefs. That is not the case in mathematics. The fundamentals laid down by the Greeks are still applicable. Important discoveries in mathematics that contradicted previous basic approaches were few and far between. Knowledge expanded enormously, new areas of research were added, unexpected applications were discovered, but today's work method is the same as that of the Greeks. Moreover, the results achieved by the Greeks and their successors throughout the generations are still relevant. No other discipline in the natural sciences can claim to have such a stable, cumulative nature. Thus, knowledge and experience play a much more important role in mathematical research than they do in other sciences. That is why we see mathematicians creating and making new discoveries at a more advanced age, as long as their enthusiasm for the subject is maintained and their health allows.

So what does a mathematician do when he arrives at his office in the morning? The answer is obvious: he drinks a cup of coffee. And then?
He drinks another cup of coffee. This description is not my idea. One of the more graphic mathematicians of the twentieth century, the Hungarian Paul Erdős (1913–1996), once defined a mathematician as a machine into which coffee is poured at one end, and at the other out come mathematical theorems. Most of a mathematician's research time is spent in thinking how to solve the problem he or she is working on. How to activate intuition varies from one mathematician to another. Some prefer to think via interaction with colleagues, others have to be alone in a perfectly silent room, others favor working while listening to classical music, and others think best while strolling around. One well-known example is that of Steve Smale of the University of California, Berkeley, a leading mathematician of the twentieth century. His many awards and prizes include the Fields Medal in 1966 and the Wolf Prize in 2007. He was given a grant so that he could devote his summer to research. The authorities discovered that he spent the summer lying on a beach in Rio de Janeiro and asked him to return the grant money he had received. Smale claimed that he was in fact working while lying on the beach, and indeed that was when he had some of his best mathematical ideas. He was able to prove his claim and convinced the committee investigating the case, which ruled in his favor. This was not just an excuse on Smale's part. I once participated in a conference at Luminy, near Marseilles in France. Steve Smale was among the participants, and he insisted that the conference timetable leave time for those who wished to do so to get inspiration on the beautiful Mediterranean beaches. Indeed, time spent on the beach did improve the quality of the lectures at the conference.

63. ON RESEARCH IN MATHEMATICS

In this and the next section I will put forward a few comments and clarifications relating to the nature of mathematical research, including some based on personal experience regarding research topics and researchers.

First, mathematics is the result of developments by many researchers. Anyone reading only the bottom line of a mathematical research study
may get the impression that mathematics was developed by a small group of geniuses (which may also be the impression gleaned from earlier chapters in this book). The truth of the matter is that the work of all those geniuses is supported by the contributions of many mathematicians, without whom they would not have made their important achievements; as time passes, however, the others are forgotten and only the leader's halo becomes brighter. Being awarded a prize or a medal also adds to the fame and glory of the individual, although in many cases the prize could also have been given to someone else equally deserving. This applies also to current research. The solution to each of the mathematical problems in the recent past, such as Fermat's theorem or Poincaré's conjecture, was the result of ongoing research carried out by many mathematicians. Despite the fact that the final stage gets all the headlines, and the mathematician who completes that stage naturally gains all the honor and praise due, the intermediate stages are often no less significant and important.

The fact that the mathematics of the Greeks is still relevant today, as is most of the mathematics developed since then, has direct implications for the subjects of research in mathematics and for mathematicians’ work methods. This situation is unlike that in other scientific disciplines, in which a research topic of several decades is likely to have become irrelevant. That is apparently why research in every other natural science is concentrated in a few major directions, whereas the range of topics recognized as worthwhile subjects of research in mathematics is much wider. The range and diversity is such that mathematicians in different areas of specialization may have difficulty understanding each other. Which is why, so it is said, one of the characteristics essential for being a mathematician is the ability to sit in a lecture in which the lecturer is describing his latest results and nod your head as if everything is clear, whereas in fact very little of what is said is understood. Obviously that is an exaggeration, but it is correct to state that the audience's understanding of a mathematics lecture is generally on the intuitive rather than the technical level. You can generally get an idea of what the lecturer is trying to achieve, what he has achieved, more or less, and possibly also the methods he uses, all in a general and intuitive way. There is hardly any chance that you will understand
the details in the lecture, unless you happen to be one of the very few mathematicians in the hall who is working in the same field. Thus, although the content of the lecture is generally presented on the logical level, the audience's understanding is usually just intuitive. A lecture to students in university, who are expected to understand the material they are studying at the logical level, is very different than a lecture in a seminar for researchers, at which most participants understand the details of only a small part of what is being said. This difference is a trap for many students beginning their research. They construe their difficulty in understanding the lecture as their weakness and find it hard to believe that the lack of technical understanding is the norm for senior faculty members.